Kinematic Space and Wormholes

TL;DR

This paper explores the geometric structure of kinematic space, revealing its intrinsic definition in embedding space and its relation to geodesics, black holes, and wormholes, providing new insights into bulk geometry reconstruction.

Contribution

It introduces a geometric construction of kinematic space independent of differential entropy, linking it to embedding space and geodesic transformations, and applies it to BTZ black holes and wormholes.

Findings

Kinematic space can be intrinsically defined in the embedding space.

SL(2,R) transformations correspond to geodesics in kinematic space.

Horizon length of BTZ black holes relates to geodesic length in kinematic space.

Abstract

The kinematic space could play a key role in constructing the bulk geometry from dual CFT. In this paper, we study the kinematic space from geometric points of view, without resorting to differential entropy. We find that the kinematic space could be intrinsically defined in the embedding space. For each oriented geodesic in the Poincar\'e disk, there is a corresponding point in the kinematic space. This point is the tip of the causal diamond of the disk whose intersection with the Poincar\'e disk determines the geodesic. In this geometric construction, the causal structure in the kinematic space can be seen clearly. Moreover, we find that every transformation in the $SL(2,\mathbb{R})$ leads to a geodesic in the kinematic space. In particular, for a hyperbolic transformation defining a BTZ black hole, it is a timelike geodesic in the kinematic space. We show that the horizon length of the static BTZ black hole could be computed by the geodesic length of corresponding points in the kinematic space. Furthermore, we discuss the fundamental regions in the kinematic space for the BTZ blackhole and multi-boundary wormholes.